I am struggling to find references or studies that explore the overall sensitivity of topological field theories as an invariant of smooth manifolds. There is the paper by Davis that explores how unitary TQFTs cannot detect smooth structure in 4-manifolds, though this is a negative result for the efficacy of TQFTs as an invariant rather than a positive one. What properties of smooth manifolds are TQFTs known to be able to detect and distinguish? This question has already asked about topological properties but was asked 11 years ago and many of the answers are as old, so perhaps the literature has advanced since then. I will also expand that question's domain to include properties pertaining to smooth structure.
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6$\begingroup$ There's a recent result of David Reutter saying that (once-)extended TFTs with values in 2Vec can't detect exotic smooth structures. $\endgroup$– Noah SnyderJun 20, 2022 at 0:22
6$\begingroup$ One dumb point, if you generalize enough, the identity TFT (with target the bordism category itself) obviously detects everything. $\endgroup$– Noah SnyderJun 20, 2022 at 0:39
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